the crude oil price, the exchange rate and
the US stock market
Jana van Leuven
Supervisor: Prof. dr. C.G.H. DiksA thesis submitted for the degree of
Bachelor of Econometrics and Operations Research December 20, 2013
1 Introduction 3
2 Theory and previous studies 5
3 Testing methodologies 7
3.1 Augmented Dickey-Fuller test . . . 7
3.2 Johansen cointegration test . . . 8
3.3 Linear Granger causality test . . . 9
3.4 Nonparametric DP test for Granger causality . . . 10
4 Description of the data 12 5 Results Granger causality tests 14 5.1 Johansen cointegration test . . . 14
5.2 Parametric and nonparametric causality testing on log-returns . . . 16
5.3 Nonparametric causality testing on the VAR residuals . . . 17
5.4 Nonparametric causality testing on GARCH-BEKK filtered VAR-residuals . . . 17
5.5 A comparison of the results with economic theory . . . 21
6 Generalized additive models 22 6.1 Estimation of a generalized additive model . . . 22
6.2 GAM representations of the causal relations . . . 23
7 Conclusion 27
1
Introduction
Precious metals such as gold and silver have received considerable attention from investors in the past few years. This is because of their potential role as hedging instrument against financial risks, such as currency depreciation and high infla-tion. Especially in the recent global financial crisis, precious metals have proven their value as a risk management tool in hedging and portfolio diversification of individual - and institutional investors. Their positive reaction to negative market shocks makes gold and silver suitable diversification instruments. Moreover, since these commodities are priced in United States dollars and traded on the futures market, they are extremely useful as a safe haven asset when the US dollar de-preciates (Baur and McDermott, 2010; Hammoudeh et al., 2010). These common purposes suggest a possible existence of interdependence between gold and silver. Therefore, an examination of this interdependence may be an interesting matter for further research.
Recent studies have revealed mixed results about the relationship between these precious metals. For example, Ciner (2001) examined the relationship between gold and silver prices and concluded that this relation has disappeared in the 1990s. Lucey and Tully (2006) however, argued that this finding might have been unwar-ranted and claim that while there are periods when the linkage is weak, overall a strong and convincing relationship conquers. Batten et al. (2013) investigated the dynamics of this relationship between gold and silver prices and found both positive and negative dependence, with the positive relation being dominant. Ham-moudeh et al. (2010) examined the co-movements among the prices of gold, silver platinum, palladium, oil and the US dollar/euro exchange rate and found a weak long-term relationship among them and a strong linkage in the short run. Thus, there has been considerable exploration on this topic so far.
However, only a few studies made use of Granger causality tests when examin-ing these linkages. Granger causality tests are valuable tools for examinexamin-ing causal relationships among time series. A linear Granger causality test is based on a coef-ficient test of a parametric, linear time series model. The nonlinear, nonparametric Granger test does not make the assumption of linearity and is therefore more ap-pealing (Bekiros and Diks, 2008). Various nonparametric tests are available. In this thesis, the test developed by Diks and Panchenko (2006) is used.
The objective of this thesis is to test for the existence of both linear and non-linear causal relations among the prices of gold, silver, oil, the United States stock market and the US dollar/euro exchange rate. Oil, as well as precious metals, is a frequently-used investment asset and historical results show co-movements among
these three commodities. The motivation for choosing the US stock market is the role of gold and silver as risk management tool in hedging and portfolio diversi-fication. The reason for choosing the US dollar/euro exchange rate is two-fold. First, the three commodities are priced in the dollar currency. Movements in the US dollar can therefore affect the prices of these goods. Second, the eurozone and the United States are large importers of oil and precious metals. Consequently, changes in the US dollar/euro exchange rate will affect the prices of these com-modities (Hammoudeh et al., 2010).
The data used for this thesis cover two periods. The first period spans from August 2003 to August 2008 and the second period spans from October 2008 to October 2013. Since the global financial crisis in 2008, gold and silver prices have risen significantly. Moreover, because precious metals have historically proven to be suitable hedging instruments against negative market shocks, it is worth explor-ing whether the financial crisis may have strengthen the causal linkages among the variables.
In this thesis, a five-step empirical framework is used for examining the rela-tionships among the five variables. First, the time series are tested on cointegra-tion. This is important because the possibility of cointegration affects the choice of model used for causality testing. If the time series are cointegrated, one should estimate a vector error correction model (VECM) rather than a vector autoregres-sive (VAR) model in first differences. The second step of the empirical framework is to apply the nonparametric Diks-Panchenko (DP) test and, while controlling for cointegration, parametric, linear Granger causality tests. Then, after filtering the time series using the VAR model or VECM, the residuals are examined by the DP test. Both the residuals of bivariate models and a full five-variate model are con-sidered. In this way, the causality between a pair of variables is studied, while taking possible effects of the other variables into account. The filtering ensures that the causality found by the DP test truly is nonlinear in nature because the lin-ear effects are already removed by the VECM or VAR model. After that, nonlinlin-ear causality is investigated after controlling for heteroskedasticity in the data using a GARCH model, again both in a bivariate and five-variate representation. In this way, remaining interdependence among the variables may be the result of causal-ity in third or higher-order moments. Finally, the nonlinear causalities among the VAR/VECM residuals are further investigated by estimating generalized additive models (GAMs). A GAM is a linear model with a predictor containing a sum of smooth functions of covariates. With the GAMs, it is possible to specify the struc-ture of the causal relations explicitly and visualize them by plotting the smooth functions.
The remainder of the thesis is organized as follows. Section 2 provides theo-retical background on the causal linkages among the five variables and the results of previous studies. Section 3 gives a description of the testing methodologies used for examining the Granger causal relations. Section 4 describes the data used. Sec-tion 5 presents the results of the Granger causality tests. In SecSec-tion 6, the GAMs are estimated and plotted. Ultimately, the conclusion follows in the last paragraph.
2
Theory and previous studies
In this section, the choice of variables for this study will be substantiated by dis-cussing the relevant theory. First, the theory about causal linkages among gold, silver and the United States stock market will be discussed, followed by the possi-ble effects of the oil price on precious metals. After that, the relationships among gold, silver, oil and the US dollar/euro exchange rate will be outlined.
As noted above, gold and silver have various common purposes. First, gold and silver are both used in the jewelry industry (Hammoudeh et al., 2010). Second, silver and gold are held as investment assets. Previous studies show significant ev-idence that these precious metals are attractive investments by themselves and can play an important role in diversifying risk (Lucey and Tully, 2006). The latter may be due to the fact that the price of gold in particular appears to react positively to negative market shocks. In other words, gold has a negative beta with the mar-ket which makes it extremely useful as hedging tool in times of stormy weather (Baur and McDermott, 2010). This can have advantages for risk diversification of portfolios held by investors. While prices of assets in their portfolios decrease in times of negative market shocks, the prices of gold and silver increase. In this way, precious metals serve as a hedge against negative market shocks, such as financial crises. It is therefore worth investigating whether there exist causal link-ages among gold, silver and movements in the United States stock market. More-over, gold and silver both can have a function as currency reserves, held by central banks as storage of value or to secure their own currency (Lucey and Tully, 2006; Baur and McDermott, 2010). These properties can lead to co-movement in gold and silver prices.
However, there are arguments against co-movement of gold and silver prices as well. Although both metals are used in industrial processes, there are differences between these usages. The majority of industrial demand for gold comes from the jewelry and dental markets, while silver is intensively used in automotive and chemicals industries (Lucey and Tully, 2006; Ghosh and Jain, 2013). Moreover, gold is more heavily used as investment asset or foreign reserve than silver (Lucey
and Tully, 2006). These contradicting arguments are reflected in the conclusions of previous studies. While Ciner (2001) found evidence of a disappearing long-term relation between gold and silver in the 1990s, Lucey and Tully (2006) claim otherwise. They state that there exist periods when the relationship is weak, but overall a stable linkage prevails. Ghosh and Jain (2013) examined the dynamics of precious metal prices in India using a linear Granger causality test and concluded that the hypothesis of non-causality could not be rejected. This study attempts to contribute to this topic by finding new empirical evidence for causal linkages between gold and silver prices.
Another interesting relationship to investigate more deeply, is the linkage be-tween oil and the precious metal prices. Similar to gold and silver, oil is considered to be an investment asset as well and included in many commodity portfolios held by individual - and institutional investors. Furthermore, oil, gold and silver are re-lated because oil enters the production process of precious metals through the use of energy inputs or fuel for transportation. An increase in the oil price can affect the production process of gold and silver negatively, which may lead to higher gold and silver prices (Hammoudeh et al., 2010). Moreover, silver is used in the auto industry which heavily utilizes oil. Lastly, increases in the oil price are often driven by inflation or financial crises. These factors also affect gold and silver prices be-cause these commodities are considered to be useful hedging instruments against inflation and negative market shocks. This can lead to co-movement of oil and the precious metals as well (Baffes, 2007).
The US dollar/euro exchange rate possibly drives the three commodities simul-taneously. This is partly due to the fact that silver, gold and oil are priced in US dollars. As noted above, investors use precious metals as a hedge against dollar depreciations to protect their wealth. If the dollar depreciates, the nominal prices of gold and silver will have a tendency to rise and therefore keep their real value. In this way, investors with dollar holdings can hedge against a falling dollar (Baur and McDermott, 2010). Another argument for co-movement of dollar exchange rates and the prices of gold, silver and oil is the demand of large importers of these com-modities. An increase in the demand for these products leads to an increase in their imports and therefore will lead to changes in the exchange rate. Or conversely; dollar fluctuations cause changes in the prices of gold, silver and oil because they are priced in US dollars (Ghosh and Jain, 2013). In this thesis, the US dollar/euro exchange rate is chosen because the three commodities are priced in US dollars and the euro and the US dollar are the world’s major reserve currencies. Moreover, the Euro area and the United States’ central banks hold the largest gold reserves and the United States and the Eurozone are both large importers of oil and precious
metals (Hammoudeh et al., 2010).
This thesis will attempt to contribute to existing studies on this topic by finding new empirical evidence on causal relationships among the five variables. How these variables relate to each other, whether there is a clear leader or driver and which relationships prevail are interesting questions to consider.
3
Testing methodologies
In this thesis, a number of testing methodologies is used to investigate the linear – and nonlinear Granger causal relations among the five variables. In this section, these testing methodologies will be presented. First, the Augmented Dickey-Fuller test for nonstationarity of time series is displayed. Next, the Johansen cointegra-tion test is discussed. The presence of cointegracointegra-tion among the five time series affects the choice of the model used for Granger causality testing. If the time series are cointegrated, a VECM should be estimated, rather than a VAR model in first differences. After that, it is described how the linear Granger causality test and the nonparametric DP test examine causal relationships among time series.
3.1 Augmented Dickey-Fuller test
The Augmented Dickey- Fuller (ADF) testing procedure requires the estimation of the following equation:
φ(L)yt= α + t, (1)
with yt = log(pt) and φ(L) = 1 − φ1L − . . . − φpLp, with L the so-called ‘lag
operator’ defined as Lkyt= yt−k. The nonstationary testing problem can be stated
as:
H0 : φ(1) = 0 (ythas a unit root),
H1 : φ(1) > 0 (ytdoes not have a unit root).
To solve this problem, it is useful to rewrite Eq. (1). Recall that for z = 1, φ(z) − φ(1)z = 0. Therefore φ(z) − φ(1)z = (1 − z)ρ(z) with ρ(z) a polynomial of order p − 1. Because ρ(0) = φ(0) − φ(1)0 = φ(0) = 1, one can write ρ(z) = 1−ρ1z−. . .−ρp−1zp−1= 1−Pp−1k=1ρkzk. With this expression for ρ(z), φ(z) can
be rewritten as φ(z) = φ(1)z +(1−z)ρ(z) = φ(1)z +(1−z)−(1−z)Pp−1
(Heij et al., 2004). Therefore, Eq. (1) becomes: {φ(1)L + (1 − L) − (1 − L) p−1 X k=1 ρkLk}yt= α + t, ⇔ ∆yt− ∆yt p−1 X k=1 ρkLk= α − φ(1)yt−1+ t, ⇔ ∆yt= α − φ(1)yt−1+ p−1 X k=1 ρk∆yt−k+ t. (2)
Then the testing problem can be restated as:
H0 : ρ = 0, H1 : ρ < 0,
where ρ = −φ(1). This can be tested with a one-sided t-test and is called the Augmented Dickey-Fuller test (Heij et al., 2004).
3.2 Johansen cointegration test
To formulate the Johansen cointegration test, consider an m-variate VAR(p) model.
Φp(L)Yt= α + t, (3)
with Yt the vector of endougenous variables and Φp(L) = Im − Φ1L − . . . −
ΦpLp, with p the number of lags. To test for cointegration, it is useful to write
the VAR(p) model as a VECM. Recall that for z = 1, Φp(z) − Φp(1)z = 0.
Thus, Φp(z) − Φp(1)z = (1 − z)Γ(z), with Γ(z) a polynomial matrix of order
p − 1. From the latter it follows that Γ(0) = Φp(0) − Φp(1)0 = Im. Therefore,
Γ(z) = Im−Pp−1j=1Γjzj and consequently Φp(L) = Φp(1)L + (1 − L)Γ(L) =
Φp(1)L + (1 − L)(Im−Pp−1j=1ΓjLj) (Heij et al., 2004). Hence, the VAR(p) model
in Eq. (3) can be rewritten as:
{Φp(1)L + (1 − L)(Im− p−1 X j=1 ΓjLj)}Yt= α + t, ⇔ Φp(1)Yt−1+ ∆Yt(Im− p−1 X j=1 ΓjLj) = α + t, ⇔ ∆Yt= α − Φp(1)Yt−1+ p−1 X j=1 Γj∆Yt−j+ t. (4)
This is called the VECM form of Eq. (3). The Johansen trace test for cointegration is an iterative test based on the rank of the matrix Π = −Φp(1). It tests H0 :
rank(Π) = r against H1 : rank(Π) ≥ r + 1, starting with r = 0. If H0 is
not rejected, then the matrix contains only zeroes and Eq. (4) reduces to a VAR(p) model of first differenced variables. There is no cointegration present in Yt. If H0is
rejected, then the new hypotheses of interest become: H0 : rank(Π) = 1 against
H1 : rank(Π) ≥ 2. If H0 is not rejected, then there is a single cointegration
relation. If H0 is rejected, one continues increasing the rank of Π. The iterative
test proceeds until the first time the test fails to reject H0. Then the number of
cointegration relations is r. The Johansen test statistic for testing H0 is given by:
LR(r) = −(n − p)
m
X
i=r+1
log(1 − ˆλi), (5)
where n is the sample size, p the lag length in Eq. (4), r the rank of Π under H0
and ˆλithe i-th eigenvalue of the matrix Π (Heij et al., 2004).
3.3 Linear Granger causality test
Granger causality testing is a technique for examining dependence relations be-tween time series. Granger (1969) states that for two stationary time series (Xt, Yt),
“Ytis causing Xtif we are better able to predict Xt using all available
informa-tion than if the informainforma-tion apart from Ythad been used” (Granger, 1969, p. 428).
In case of the linear Granger causality test, one imposes a parametric test on the coefficients of a VAR model. Specifically, in case of two time series {Xt} and
{Yt} ∼ I(1), the bivariate VAR model is given by:
∆Xt= ξ1+ Φp(L)∆Xt+ Θp(L)∆Yt+ ∆X,t,
∆Yt= ξ2+ Ψp(L)∆Xt+ Ωp(L)∆Yt+ ∆Y,t,
(6)
where ∆Xtand ∆Ytare stationary time series of the first differences of {Xt} and
{Yt} and Φp(L), Θp(L), Ψp(L) and Ωp(L) are polynomials in the lag operator
L. The error terms are independent and identically distributed variables with zero mean and constant variance. The parametric test whether X is Granger causing Y is a test of the restriction that all coefficients of the polynomial Ψp(L) are zero. If
the test statistic rejects this joint restriction, then the null hypothesis of no Granger causality is rejected and X is Granger causing Y . Testing whether Y is Granger causing X boils down to a test of the restriction that all coefficients of Θp(L)
linear causality testing is carried out using the Wald version of the linear Granger causality test (Heij et al., 2004).
3.4 Nonparametric DP test for Granger causality
In addition to the parametric test, one can perform a nonparametric test for Granger causality. To describe the procedure of the nonparametric DP test, it is useful to provide a definition of Granger causality. As noted above, Granger causality states that, for a stationary bivariate process, Xt is a Granger cause of Yt if past and
current values of X possess additional information on future values of Y that is not included in past and current values of Y alone (Diks and Panchenko, 2006). Let FX,tand FY,trepresent the information sets comprised of past observations of
Xtand Ytrespectively, and let ‘∼’ denote equivalence in distribution. Then Xtis
a Granger cause of Ytif, for some k ≥ 1:
(Yt+1, . . . , Yt+k)|(FX,t, FY,t) 6∼ (Yt+1, . . . , Yt+k)|FY,t. (7)
In contrast to the linear Granger causality test, which assumes a parametric, linear time series model, this definition does not imply any modelling assumptions and is therefore often referred to as general or nonlinear Granger causality. Note that Eq. (7) refers to conditional distributions given an infinite number of past observations, captured in the information sets FX,tand FY,t. However, tests are usually limited
to finite orders (Diks and Panchenko, 2006).
In testing for Granger non-causality, the aim is to find evidence against the null hypothesis:
H0: {Xt} is not Granger causing {Yt}, (8)
with Granger causality defined in Eq. (7). In practice, k = 1 is most often used, in which case non-causality testing boils down to comparing the conditional dis-tribution of Yt+1 with and without past and current values of {Xt} and conclude
if these distributions are different or not. Therefore, the null hypothesis in Eq. (8) can be rewritten as:
Yt+1|(XtlX; Y lY t ) ∼ Yt+1|YtlY, (9) where XlX t = (Xt−lX+1, . . . , Xt) and Y lY
t = (Yt−lY+1, . . . , Yt) are the delay
vectors, with lX and lY finite lags. To keep the notation compact, define Zt= Yt+1.
Moreover, often the time index is dropped and lX = lY = 1 is taken (Diks and
Panchenko, 2006). Hence, under the null hypothesis in Eq. (9), the conditional distribution of Z given (X, Y ) = (x, y) is the same as the conditional distribution
of Z given Y = y. Therefore, the null hypothesis becomes:
fZ|X,Y(z|x, y) = fZ|Y(z|y). (10)
There exist various nonparametric test statistics for the Granger non-causality hypothesis. In this thesis, the test statistic proposed by Diks and Panchenko (2006) is used. To motivate the test statistic, it is useful to reformulate the null hypothesis in Eq. (10) in terms of ratios of joint distributions:
fX,Y,Z(x, y, z) fX,Y(x, y) = fY,Z(y, z) fY(y) ⇔ fX,Y,Z(x, y, z) fY(y) = fX,Y(x, y) fY(y) fY,Z(y, z) fY(y) . (11)
As noted by Diks and Panchenko (2006), this reformulated null hypothesis implies:
q ≡ E fX,Y,Z(x, y, z) fY(y) −fX,Y(x, y) fY(y) fY,Z(y, z) fY(y) g(X, Y, Z) = 0, (12)
with g(x, y, z) a weight function. In particular, Diks and Panchenko (2006) chose g(x, y, z) = fY2(y). Therefore, the null hypothesis of Granger non-causality be-comes:
q ≡ E[fX,Y,Z(X, Y, Z)fY(Y ) − fX,Y(X, Y )fY,Z(Y, Z)] = 0. (13)
Let W = (X, Y, Z) denote a random vector with the invariant distribution of (XlX
t , Y
lY
t , Yt+1) and ˆfW(Wi) a local density estimator of the random vector
W at Wi defined by ˆfW(Wi) = (2)dW(n − 1)−1Pj,j6=iIijW where dW is the
dimension of W and IijW = I(||Wi− Wj|| < nwith I(·) the indicator function
and n the bandwidth, depending on the sample size n. Given this estimator, the
DP test statistic as estimator of q becomes:
Tn() =
(n − 1) n(n − 2)
X
i
( ˆfX,Y,Z(Xi, Yi, Zi) ˆfY(Yi) − ˆfX,Y(Xi, Yi) ˆfY,Z(Yi, Zi)).
(14) Diks and Panchenko (2006) prove that for dX = dY = dZ = 1 this test is
consis-tent if the bandwidth depends on the sample size as n = Cn−β with C > 0 and
β ∈ (14,13). Moreover, Diks and Panchenko (2006) prove that in the absence of dependence between the vectors Wi, this test statistic is asymptotically normally
distributed. Furthermore, under certain mixing conditions, Tn(n) in Eq. (14)
sat-isfies: √ nTn(n) − q Sn d − → N (0, 1), (15)
where−→ denotes convergence in distribution and Sd nis an estimator of the
asymp-totic variance of Tn(n) (Diks and Panchenko, 2006). Consequently, testing for
Granger non-causality boils down to testing whether Tn(n) in Eq. (14) is
signifi-cantly larger than zero. In that case the null hypothesis of non-causality is rejected.
4
Description of the data
The data consist of time series of daily gold –, silver – and oil prices, daily US dol-lar/euro exchange rates and daily observations of the Standard and Poor’s 500 index (hereafter S&P 500 index). The daily gold prices are issued by Handy&Harman, a North American manufacturing company of, among others, precious metals. The gold price is measured in US dollars per troy ounce. Silver is traded at the New York Mercantile Exchange and its price is also measured in US dollars per troy ounce. For the crude oil data, the daily prices of West Texas Intermediate (WTI), a type of crude oil and an important benchmark of crude oil pricing are used. WTI is delivered at the trading hub in Cushing, Oklahoma and priced in US dollars per barrel. The main reason for choosing these commodity prices in particular is be-cause they are all issued in North America and therefore have the same time zone and currency. The daily US dollar/euro exchange rate data are noon buying rates in New York City and measured as the US dollar value of one euro. Thus, an in-crease in the exchange rate implies a devaluation of the US dollar. The S&P 500 index is used as a measurement of the United States stock market. Additionally, all variables are divided by the Consumer Price Index (CPI). This conversion to real prices is important because inflation may influence the variables simultaneously. Consequently, not correcting for inflation can result in spurious causal linkages. To avoid this, each time series is divided by the CPI.
The data cover two periods. The first period spans from August 29 2003 to August 29 2008 and the second period spans from October 1 2008 to October 1 2013. The separation of the sample corresponds roughly to a pre-global financial crisis period and a period that commences immediately after the start of the global financial crisis in September 2008, when Lehman Brothers declared bankruptcy. Fig. 1 displays the time series of the three commodities, the S&P 500 index and the exchange rate in both periods. The following notation is used: “Gold”, “Slvr” and “Oil” are the prices of the three commodities and “S&P” and “XR” are the S&P 500 index and the US dollar/euro exchange rate respectively. All variables are in natural logarithms. Observing the time series, the start of the global financial crisis is clearly visible by a downfall in all five time series. The oil price dropped heavily, as well as the S&P 500 index and the exchange rate. The prices of the precious metals
only experienced a small drop and started increasing again after September 2008. The exchange rate shows increased volatility in the second period. One suspects that all five variables are nonstationary in both periods. Therefore, a formal test for nonstationarity is performed based on the Augmented Dickey-Fuller (ADF) test.
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 Gold -4.0 -3.6 -3.2 -2.8 -2.4 -2.0 -1.6 -1.2 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 Slvr -2.0 -1.6 -1.2 -0.8 -0.4 0.0 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 Oil 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 S&P -5.25 -5.20 -5.15 -5.10 -5.05 -5.00 -4.95 -4.90 -4.85 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 XR
Figure 1: The time series of the commodity prices, the S&P 500 index and the exchange rate in PI: 8/29/2003 -8/29/2008 and PII: 10/1/2008 - 10/1/2013
The ADF test is performed on the logarithmic levels and logarithmic returns of the five time series. Specifically, the log-returns are defined as rt = log(Pt) −
log(Pt−1), where Pt is the real price on day t. The lag lengths p in Eq. (2) are
chosen based on the Schwartz Information Criterion (SIC). Table 1 displays the results of the ADF test. The first column reports the log-levels and log-returns. The lag lengths are displayed in parenthesis. The number of lags is zero in all cases except for the S&P 500 index, in which case it is one. The second and third column report the p-values of the ADF test in period one and period two
respectively. All five variables appear to be nonstationary in levels and stationary in returns. Therefore, one can conclude that the variables are integrated of order one.
Variables p-values (PI) p-values (PII)
Gold (0) 0.8185 0.5598 rGold(0) 0.0000 0.0000 Slvr (0) 0.4788 0.4129 rSlvr(0) 0.0000 0.0000 Oil (0) 0.7068 0.2386 rOil(0) 0.0000 0.0000 S&P (1) 0.1812 0.7539 rS&P(0) 0.0000 0.0000 XR (0) 0.1959 0.2511 rXR(0) 0.0000 0.0000
The variables are log-levels and log-returns and the number of lags in parenthesis is chosen using the Schwartz Information Criterium. PI refers to period one: 8/29/2003 8/29/2008 and PII refers to period two: 10/1/2008 -10/1/2013.
Table 1: Results ADF test
5
Results Granger causality tests
In this section, the results of the parametric – and nonparametric Granger causality tests will be presented. The testing procedure covers a four-step framework for examining the causal linkages among the five variables. First, the time series are tested on cointegration. Then, the nonparametric DP test and, while controlling for cointegration, linear Granger causality tests are applied to the log-returns. The lin-ear test is applied to both bivariate models and a full five-variate model. In this way, Granger causality between a pair of variables is examined, while taking possible effects of the other variables into account. After that, the residuals of the VECMs or VAR models are examined by the DP test. Both the residuals of the bivariate models and the full five-variate model will be considered. The filtering ensures that the remaining causality found by the DP test is strictly nonlinear in nature. Finally, the data are filtered using a GARCH model, again both in a bivariate and five-variate representation. Remaining interdependence among the variables may then be the result of causality between third or higher-order moments.
5.1 Johansen cointegration test
The presence of cointegration between time series affects the choice of the model used for Granger causality testing. If the time series are cointegrated, a VECM
should be specified. If not, causality testing should be based on VAR model of log-returns. Therefore, testing for cointegration is necessary before commencing with the Granger causality tests.
The results of the Johansen cointegration test on the time series are reported in Table 2. All variables are log-levels and the lag lengths p are chosen based on the Schwartz Information Criterium (SIC). The reported numbers are p-values cor-responding to the null hypothesis: rank(Π) = 0. It appears that in all cases the Johansen test fails to reject the null of no cointegration except for Slvr and Oil in PII. Thus, the test suggests that these two variables are cointegrated. However, one should not immediately conclude that the time series are indeed cointegrated, only judging from the test results. The rejection of the null may, for example, be the result of model misspecifications. Judging from economic theory, it seems more reasonable that Slvr and Oil are not cointegrated because these are different com-modities. Moreover, if the results of the cointegration test are doubtful, it is more safe to choose the VAR model instead of the VECM. When specifying a VECM, one imposes an extra restriction on the data in the form of an error correction term in the model specification. It is therefore safer to choose the more general VAR model when cointegration is doubtful. Hence, we proceed assuming that none of the variables are cointegrated. This means that for the application of the causality tests, for all pairs and for the five-variate implementation, a VAR model of log-returns should be estimated.
Variables PI PII
lags p-value lags p-value
Gold Slvr 3 0.4789 2 0.4803 Gold Oil 2 0.6286 1 0.0568 Gold S&P 2 0.4238 1 0.8605 Gold XR 1 0.4654 1 0.5155 Slvr Oil 1 0.7777 1 0.0388 Slvr S&P 2 0.4492 2 0.9389 Slvr XR 1 0.6508 1 0.6210 Oil S&P 2 0.3135 2 0.2614 Oil XR 1 0.6997 1 0.4748 S&P XR 2 0.4648 2 0.5457
All five variables 2 0.3104 2 0.2358
All variables are log-levels and the number of lags is selected using the SIC. The reported numbers are p-values of the null hypothesis: rank(Π) = 0. PI refers to period one: 8/29/2003 - 8/29/2008 and PII refers to period two: 10/1/2008 - 10/1/2013.
5.2 Parametric and nonparametric causality testing on log-returns
The linear Granger causality test is a parametric test on the coefficients of a VAR model. Because the time series are all I(1), the linear Granger causality tests are based on VAR models of log-returns. The lag-lengths of the models are chosen based on the Schwartz Information Criterium. For each pair, the optimal lag length is one in both periods, except for Gold and Slvr in PI, for which case it is two. In the five-variate representation, the optimal lag length is one in both periods. The results of the parametric linear causality tests are presented in Table 3. One can observe the following: In the pairwise implementation, Slvr, Oil, S&P and XR linearly Granger cause Gold with small differences in the degree of significance between PI and PII. Gold only Granger causes S&P and only in PI. Furthermore, there are strong unidirectional relationships Slvr → XR and S&P → XR in both periods and a bi-directional relation Slvr ↔ S&P in PI. Moreover, S&P presents a significant unidirectional relationsip S&P → Slvr in PII. Finally, Oil presents strong unidirectional relationships Oil → S&P and Oil → XR in PI but not in PII. In PII however, opposite unidirectional relations appear to be significant. In the five-variate implementation, a number of significant relationships disappeared. For instance, Oil → Gold, S&P → Gold in PII and Slvr → S&P in PI have vanished. Moreover, Gold and XR as well as Slvr and Oil seem to lack any causal relation. The relationships between Slvr and S&P, Slvr and XR in PII and S&P and XR in PI are now bi-directional.
For the Diks-Panchenko test, the lag lengths lx, ly and the bandwidth n are
Variables Linear Granger causality X Y Pairwise Five-variate
X → Y Y → X X → Y Y → X PI PII PI PII PI PII PI PII
Gold Slvr *** *** *** *** Gold Oil *** *** *** Gold S&P *** *** ** ** ** Gold XR *** ** Slvr Oil ** Slvr S&P ** *** *** ** *** *** Slvr XR *** *** *** ** Oil S&P *** *** ** *** Oil XR *** ** *** *** S&P XR *** *** *** *** **
All variables are log-returns. (**)/(***) denotes p-value significance at 5% and 1%.
set to one. Bandwidth values smaller (larger) than one generally resulted in larger (smaller) p-values. Choosing a smaller bandwidth reduces the risk of over-rejection of the null hypothesis but at the same time leads to less significant causal relations. Therefore, the bandwidth is set to one. The results of the nonparametric test on the log-returns are presented in Tables 4 and 5. Interestingly, there now is strong evidence of bi-directional relationships Gold ↔ Slvr and Gold ↔ Oil in PII. The linear relation Oil → Gold in PI disappeared. Moreover, any previously detected linear causal relations among Oil, S&P and XR in PI have vanished. However, strong bi-directional causal relations appear to be among them in PII. Again, Gold and XR do not seem to relate in any of both periods. The causal relation Slvr ↔ XR in PII reduced to a unidirectional relation XR → Slvr. Highly significant causal relationships Oil → Slvr and XR → S&P now emerged in PII. Finally, comparing PI and PII, the DP test detected more causal relationships among the five variables in PII.
5.3 Nonparametric causality testing on the VAR residuals
Although the results of the nonparametric test, given in the previous section, imply causal linkages among the five variables, the DP test should be reapplied to the VAR residuals to determine the nature of the causality. Any remaining causality among the VAR residuals found by the DP test is strictly nonlinear in nature be-cause the VAR model filtered out the linear effects. The results of the nonlinear causality tests on the VAR residuals are presented in Tables 4 and 5.
Comparing the results of the DP test on the log-returns and the VAR residuals in Table 4, the test results are almost identical except for the absence now of the causal relation Gold → Slvr and a less significant relation Oil → Gold in PII. In the five-variate representation, the causal relationships Gold → Slvr, Gold → Oil and XR → Slvr in PII disappeared. The relationship Gold → S&P in PI emerged, while the relation S&P → Gold in the same period decreased in significance. The DP test detected more causal relations in PII than in PI, both for the log-returns and the VAR residuals. In case of the linear Granger causality test results, this distinction between the two periods is not visible.
5.4 Nonparametric causality testing on GARCH-BEKK filtered
VAR-residuals
By applying the DP test on the VAR residuals, it was possible to distinguish be-tween linear and nonlinear Granger causality. By filtering the VAR residuals with a multivariate GARCH model, it is possible to investigate whether the nonlinear
Variables Nonlinear Granger causality
X Y Log-returns VAR residuals GARCH-BEKK residuals X → Y Y → X X → Y Y → X X → Y Y → X PI PII PI PII PI PII PI PII PI PII PI PII Gold Slvr ** *** *** *** *** *** ** Gold Oil ** *** ** ** Gold S&P *** *** *** *** *** Gold XR Slvr Oil *** *** Slvr S&P ** ** *** ** ** *** ** ** ** Slvr XR ** ** Oil S&P *** *** *** *** Oil XR *** ** *** ** S&P XR *** *** *** ***
(**)/(***) denotes p-value significance at 5% and 1%. The first four columns display the results of the DP test on the log-returns. The second four columns show the results of the DP test on the VAR residuals of the bivariate models and the last four columns show the results of the DP test on the residuals after GARCH-BEKK filtering.
Table 4: Results nonparametric DP test (Pairwise)
Variables Nonlinear Granger causality
X Y Log-returns VAR residuals GARCH-BEKK residuals X → Y Y → X X → Y Y → X X → Y Y → X PI PII PI PII PI PII PI PII PI PII PI PII Gold Slvr ** *** *** *** *** *** ** Gold Oil ** *** *** Gold S&P *** *** ** ** *** Gold XR Slvr Oil *** *** Slvr S&P ** ** *** ** ** *** ** Slvr XR ** Oil S&P *** *** *** *** Oil XR *** ** *** ** ** S&P XR *** *** *** ***
(**)/(***) denotes p-value significance at 5% and 1%. The first four columns display the results of the DP test on the log-returns. The second four columns show the results of the DP test on the VAR residuals of the five-variate models and the last four columns show the results of the DP test on the residuals after GARCH-BEKK filtering.
Granger causality lies in the conditional variances. If this is the case, then causality will be reduced when the appropriate GARCH model is fitted to the VAR residu-als. However, if the DP test rejects the null hypothesis of no causality after filtering, this can also mean that the GARCH model is incorrectly specified. Therefore, one should interpreter the results with care.
There are various possible GARCH models available for modeling the condi-tional covariance matrix. Consider the following m-variate model:
Yt= α + ηt, (16)
where Ytis a vector containing the VAR residuals and ηt|Yt−1∼ N (0, Ht). In this
thesis, the diagonal-BEKK(1,1,1) model is chosen for modeling the conditional covariance of ηt. This implies the following specification of Ht:
Ht= C0C + A0ηt−1ηt−10 A + G0Ht−1G, (17)
where C, A and G are m × m matrices and C is upper triangular. Additionally, because this is a diagonal-BEKK model, A and G are diagonal matrices (Bauwens et al., 2006). The standardized residuals ˆvt, used for the nonparametric causality
testing analysis, are obtained by the following transformation: ˆvt = ˆH
−1
2 t ηˆt.
Ta-bles 4 and 5 show the results of the DP test after GARCH-BEKK filtering. One can conclude that the nonlinear causality among the variables is strongly reduced after filtering. Only a few causal relations remain significant, both in the pairwise and five-variate implementation. This indicates that the nonlinear causality for a large part lies in the conditional covariances and variances of the time series. How-ever, some nonlinear causal linkages remain even after filtering. In particular, the pairwise implementation shows the unidirectional relations Slvr → Gold in both periods and S&P → Gold in PI. Additionally, the bi-directional relation Slvr ↔ S&P in PI remains significant. The statistical significance of the linkage S&P → Slvr in PII is weaker after filtering. In the five-variate case, Slvr → Gold in PI is still strongly significant. The unidirectional causal relations Slvr → Gold, Oil → XR in PII and S&P → Slvr in PI are weaker but still present. The remaining causal relations possibly imply that volatility effects are not the only sources induc-ing nonlinear causality and causality in higher-order moments may be a significant factor of the remaining interdependence.
The nonlinear causal relationships among the VAR residuals before and after filtering are graphically displayed in Figs. 2 and 3 respectively. Causality at 1% significance (***) is denoted by a thick arrow and causality at 5% significance (**) is denoted by a thinner arrow.
Figure 2: Graphical representation of the causal relations among the VAR residuals before GARCH-BEKK filtering. The thick arrows denote a significance at 1%, the thin arrows a significance at 5%.
Figure 3: Graphical representation of the causal relations among the VAR residuals after GARCH-BEKK filtering. The thick arrows denote a significance at 1%, the thin arrows a significance at 5%.
5.5 A comparison of the results with economic theory
It is interesting to exame whether the empirical results in this thesis and economic theory coincide. In this comparison, the focus is on the relation between gold and silver and between gold and silver on one hand and the oil price, the market index and the exchange rate on the other hand. The reason for this choice is that the initial goal was to investigate the relation between gold and silver while taking into account the effects of the other three variables.
Referring back to economic theory, one can notice a number of interesting test results. First, it was expected that gold would Granger cause silver. The increased use of gold as a hedge against financial risks urged investors to start using other precious metals as risk management tools and portfolio diversification instruments as well. Likewise, the trading volume of gold is larger than that of silver. The test results however, indicate little evidence for this development. While the DP test found a bi-directional relationship between the returns of gold and silver in PII, the linear test and the nonlinear test on the VAR residuals could only find evidence for silver Granger causing gold in both periods. Second, all tests except the bivariate implementation of the linear test, fail to reveal any causal linkages between the gold price and the US dollar/euro exchange rate. Economic theory suggests interdependence between the two variables because gold is frequently used as a safe haven asset when the US dollar is expected to depreciate. The fact that the Granger causality tests fail to find empirical evidence for this relationship does not mean that this relation does not exist. The relationship is just not dominant compared to other relations involving the dollar exchange rate. Both the US dollar and the euro experienced a turbulent period over the past few years and there are many other factors that drive these two currencies.
Gold, silver and oil are frequently-used investment assets and included in many commodity portfolios of individual – and institutional investors. Moreover, the role of gold and silver as risk management tools suggests linkages among the prices of gold, silver and the US stock market. The DP test shows a unidirectional causal relation from the S&P 500 index to gold in both periods and the linear test shows a bi-directional relation in PI. The test results regarding silver and the market index are also consistent with the expectations by showing a bi-directional nonlinear re-lation in PI and a bi-directional linear rere-lation in PII. Lastly, the DP test suggests a bi-directional relationship between oil and the S&P 500 index in PII, while the linear test point towards a unidirectional relation from oil to the S&P 500 index in PI and an opposite unidirectional relation in PII.
pre-cious metals through the use of energy inputs or fuel for transportation. Moreover, oil and silver are both heavily utilized in the auto industry. The test results show a linear relation from oil to gold in PI and a nonlinear relation from oil to gold in PII. The DP test shows a bi-directional causal relation between the returns of gold and oil in PII. Moreover, the DP test indicates a unidirectional relation from oil to silver in PII. Therefore, it can be concluded that the empirical results support the theory.
Thus, in general the empirical results and the economic theory concur. Incon-sistent results are the lack of interdependence between gold and the exchange rate and the direction of the relationship between silver and gold.
6
Generalized additive models
In the previous section, the Granger causality tests have indicated causal relation-ships among the five variables. However, the test results tell us nothing about the structure of these causal relations. In this section, the Granger causalities are inves-tigated in more detail by estimating generalized additive models (GAMs). A GAM is a linear model with a predictor containing a sum of smooth functions of covari-ates. Rather than specifying an elaborated parametric relation, a GAM allows for more flexibility by specifying the model only in terms of smooth functions (Wood, 2006). The GAM can be used to specify and visualize the nonlinear causal rela-tions found by the Granger causality tests. In this thesis, the general structure of the GAM is as follows:
yt= f1(yt−1) + f2(xt−1) + t, (18)
where ytand xtare stationary time series, fj are smooth functions and t are
in-dependent and identically distributed N (0, σ2) random variables. If, for example, the Granger causality test found the nonlinear relation x → y, then by estimat-ing the model in Eq. (18), the function ˆf2(xt−1) will give an idea of the form of
interdependence between x and y.
The first part of this section gives an illustration how GAMs can be estimated. In the second part, different GAMs will be specified to model the Granger interde-pendencies found in Section 5.
6.1 Estimation of a generalized additive model
Consider the GAM in Eq. (18). To estimate this model, the first step is to specify a basis for each smooth function fj, with j = 1, 2. If bjiis the i-th basis function of
fjthen the smooth functions can be represented as: fj(zj) = qj X i=1 βjibji(zj), j = 1, 2, (19)
where (z1, z2) = (yt−1, xt−1) and βji are coefficients of the smooth functions
which need to be estimated. There is a variety of possible bases available. The bases used in this study are thin plate regression splines (Wood, 2006).
The second step is to determine the degree of smoothness for each smooth function fj. This controls the shape of the function and how well it fits the data.
In determining the shape of fj, it is needed to make a tradeoff between model fit
and model smoothness. This tradeoff is controlled by the smoothing parameters λj. Large values of λj lead to a large penalty on the smoothness of fj and λj = 0
leads to an un-penalized estimation of fj. This means that if λj is too high, fj fits
the data poorly, while too low values for λj lead to a wiggly function. Therefore,
the problem of estimating the degree of smoothness of the model boils down to estimating the smoothing parameters λj. One method for choosing the optimal
values of λ = (λ1, λ2) is generalized cross validation (GCV). One chooses the λ
that minimizes the GCV score (Wood, 2006).
Given a basis and a measurement for the smoothness for each smooth function, the next step is to estimate the coefficients βji of the GAM. GAMs are usually
estimated by penalized likelihood maximization. A penalty for overly wiggly esti-mates of fj is added to the maximization objective in order to control the
smooth-ness of the model. Given values for the smoothing parameters λj, the penalized
likelihood lp(β) is maximized to find the estimations of the coefficients ˆβ. This is
done by penalized iteratively re-weighted least squares (Wood, 2006).
Thus, estimating a GAM starts with a choice of basis for each smooth func-tion fj. Next, the smoothing parameters λj are estimated using cross validation.
Then the model coefficients βjiare estimated by maximizing a penalized likelihood
function that suppresses overly curvy estimates of fjterms.
6.2 GAM representations of the causal relations
In Section 5, nonlinear causal relations were found among the five variables by applying the DP test on the VAR residuals. In this section, these relationships are investigated in more detail by estimating GAMs. With a GAM, it is possible to model the nonlinear Granger interdependencies and by plotting the smooth func-tions ˆf2, the structure of the nonlinear relationships can be visualized.
because it is preferred to model the nonlinear Granger causalities. The DP test found sixteen nonlinear causal relations (see Table 5). The model in Eq. (18) is estimated for all sixteen pairs and the results are displayed in Table 6. The first three columns display the period and the time series considered. The last two columns report the p-values of the smooth functions f1and f2. The p-values should
be interpreted with care because they are typically lower than they should be. The reason for this is that the smoothing parameters βji are unknown and therefore
needed to be estimated. The uncertainty of the parameters has been neglected in the reference distributions used for testing, resulting in distributions that are too narrow. Consequently, the test rejects the null too readily (Wood, 2006). For this reason, only the smooth functions with p-values lower than 1% are considered to be significant. Observing Table 6, this means that six smooth functions f2 are
significant. These six relations will be discussed.
Period Time Series p-values
yt xt f1(yt−1) f2(xt−1)
PI Gold Slvr 0.00295*** 6.23e-07*** PII Gold Slvr 0.00250*** 3.61e-05*** PII Gold Oil 0.00089*** 0.566610 PI S&P Gold 0.31200 0.28000 PI Gold S&P 0.04093 0.00959*** PII Gold S&P 0.00121*** 0.61715 PII Slvr Oil 0.00140*** 0.44590 PI S&P Slvr 0.06560 0.97060 PI Slvr S&P 0.00033*** 0.02043 PII Slvr S&P 0.00324*** 0.21270 PII S&P Oil 0.38177 0.00464*** PII Oil S&P 0.76000 2.05e-05***
PII XR Oil 0.36500 0.24300
PII Oil XR 0.78501 0.00631***
PII XR S&P 0.52580 0.03990 PII S&P XR 0.04960 0.11380
The time series are residuals of the five-variate VAR models. The reported numbers are p-values of the smooth
functions in the GAM yt= f1(yt−1) + f2(xt−1). (***) denotes significance at 1%. PI refers to period one:
8/29/2003 - 8/29/2008 and PII refers to period two: 10/1/2008 - 10/1/2013.
Table 6: Estimation results GAMs
The first relationship of interest is Slvr → Gold in PI and PII. The plots of the smooth functions of Slvr with 95% confidence limits are displayed in Figs. 4 and 5. These confidence limits however, assume independence between the time series Goldt, Goldt−1and Slvrt−1. Because these are most likely not independent, one
should interpret these confidence limits carefully. The same goes for the confidence limits of the other smooth functions. From Figs. 4 and 5, one can immediately see that the structure of the nonlinear relation Slvr → Gold is different in PI and PII.
Next, the causal relation S&P → Gold in PI is considered. The smooth function of S&P is displayed in Fig. 6. The remaining three GAMs with significant smooth functions ˆf2correspond to the relations Oil → S&P, S&P → Oil and XR → Oil in
PII. The associated smooth functions are displayed in Figs. 7, 8 and 9 respectively. All smooth functions indicate a nonlinear structure.
−0.15 −0.10 −0.05 0.00 0.05 −0.02 −0.01 0.00 0.01 0.02 0.03 0.04 Slvr_{t−1} E(Gold_t)
Figure 4: Plot of ˆf2(Slvrt−1) with 95%
confi-dence limits (PI)
−0.15 −0.10 −0.05 0.00 0.05 −0.04 −0.02 0.00 0.02 0.04 Slvr_{t−1} E(Gold_t)
Figure 5: Plot of ˆf2(Slvrt−1) with 95%
confi-dence limits (PII)
−0.02 0.00 0.02 0.04 −0.03 −0.02 −0.01 0.00 0.01 0.02 S&P_{t−1} E(Gold_t)
Figure 6: Plot of ˆf2(S&Pt−1) with 95%
confi-dence limits (PI)
−0.10 −0.05 0.00 0.05 0.10 −0.02 −0.01 0.00 0.01 0.02 0.03 Oil_{t−1} E(S&P_t)
Figure 7: Plot of ˆf2(Oilt−1) with 95%
−0.10 −0.05 0.00 0.05 0.10 −0.10 −0.06 −0.02 0.00 0.02 0.04 S&P_{t−1} E(Oil_t)
Figure 8: Plot of ˆf2(S&Pt−1) with 95%
confi-dence limits (PII)
−0.02 0.00 0.02 0.04 −0.04 −0.02 0.00 0.02 XR_{t−1} E(Oil_t)
Figure 9: Plot of ˆf2(XRt−1) with 95%
confi-dence limits (PII)
Until now, only GAMs with two smooth functions are considered. However, the GAM can be extended to, for example, a model with three smooth functions. Because both silver and the S&P 500 index are Granger causing gold in PI, it is interesting to estimate a GAM with three smooth functions and to examine if the smooth functions for Slvr and S&P in this model are different from the smooth functions in the separate models. Thus, the following GAM is estimated:
Goldt= f1(Goldt−1) + f2(Slvrt−1) + f3(S&Pt−1) + t. (20)
The p-values of f1, f2and f3are 0.00358, 3.53e-07 and 0.00902 respectively. The
plots of the smooth functions f2 and f3 are displayed in Figs. 10 and 11
respec-tively. Comparing the p-values of this model with the separate models, no large differences are noticeable. Comparing the plots of Figs. 10 and 11 with Figs. 4 and 6, one can conclude that estimating a model with three smooth functions or estimating two separate models leads to almost identical results. The same can be carried out for the exchange rate, the S&P 500 index and the oil price. Because both the exchange rate and the S&P 500 index are Granger causing oil in PII, the following model is estimated:
Oilt= f1(Oilt−1) + f2(S&Pt−1) + f3(XRt−1) + t. (21)
Both f2and f3are significant with p-values of 2.91e-05 and 0.00743 respectively.
The plots are displayed in Figs. 12 and 13. Comparing the results with Figs. 8 and 9, it can be concluded that the smooth functions of the model in Eq. (21) are similar to the smooth functions of the separate models.
−0.15 −0.10 −0.05 0.00 0.05 −0.02 0.00 0.02 0.04 Slvr_{t−1} E(Gold_t)
Figure 10: Plot of ˆf2(Slvrt−1) with 95%
confi-dence limits (PI)
−0.02 0.00 0.02 0.04 −0.02 0.00 0.02 0.04 S&P_{t−1} E(Gold_t)
Figure 11: Plot of ˆf3(S&Pt−1) with 95%
confi-dence limits (PII)
−0.10 −0.05 0.00 0.05 0.10 −0.10 −0.06 −0.02 0.00 0.02 S&P_{t−1} E(Oil_t)
Figure 12: Plot of ˆf2(S&Pt−1) with 95%
confi-dence limits (PII)
−0.02 0.00 0.02 0.04 −0.10 −0.06 −0.02 0.00 0.02 XR_{t−1} E(Oil_t)
Figure 13: Plot of ˆf3(XRt−1) with 95%
confi-dence limits (PII)
Thus, estimating GAMs can be useful if one is interested in examining the structure of the Granger causalities. By plotting the smooth functions, the causal relationships can be visualized.
7
Conclusion
In this thesis the linear and nonlinear causal relationships among the prices of gold, silver, oil, the United States stock market and the US dollar/euro exchange rate were investigated. The data covered two periods. The first period spanned from
August 29 2003 to August 29 2008, the second period spanned from October 1 2008 to October 1 2013. This segmentation corresponds roughly to a pre-global financial crisis period and a period that commences immediately after the start of the global financial crisis. The five time series were corrected for inflation by dividing them by the consumer price index to avoid spurious causal linkages.
The causal relationships among the five variables were investigated with para-metric – and nonparapara-metric Granger causality tests. The parapara-metric Granger causal-ity test was performed on bivariate – and five-variate VAR models. The nonpara-metric DP test was performed on the log-returns and the residuals of the VAR models. Moreover, the DP test was applied to the VAR residuals after filtering them with a diagonal-BEKK model.
Several conclusions could be drawn from the results of the causality tests. In particular, it was shown that the pairwise VAR modeling suggested unidirectional linear Granger causality from silver to gold in both periods. The DP test pointed to-wards a unidirectional relation from silver to gold in PI and a bi-directional relation between silver and gold in PII. The application of the DP test on the VAR residu-als resulted in a unidirectional relation from silver to gold in both periods. Thus, surprisingly little evidence for gold Granger causing silver has stemmed from the test results. Economic theory suggested that gold and silver relate because they are both used as investment assets and included in many commodity portfolios of investors. Moreover, because these precious metals appear to react positively to negative market shocks, they play an important role in diversifying portfolio risk. Gold is seen as the starting point for the movement towards utilizing precious metals as risk management tools in hedging and portfolio diversification. It was therefore expected that gold would Granger cause silver. However a majority of the Granger causality tests indicated an opposite relationship between silver and gold.
This role of the precious metals as risk management tools also suggested link-ages among the prices of gold, silver and the US stock market. The five-variate implementation of the linear test indicated a bi-directional Granger causal relation between the gold price and the S&P 500 index in PI. The application of the DP test on the log-returns resulted in a unidirectional relation from the S&P 500 index to gold in both periods. A unidirectional nonlinear relation from the S&P 500 in-dex to gold in PI and a bi-directional nonlinear relation between the two in PI was found by applying the DP test on the VAR residuals. Moreover, empirical evidence was found for the existence of causal linkages between silver and the S&P 500 in-dex. In particular, it was shown that the five-variate VAR modeling suggested a strong unidirectional linear relation from the S&P 500 index to silver in PI and a
bi-directional linear Granger causality between the two variables in PII. The DP test found evidence for a bi-directional relationship between silver and the S&P 500 index in PI and a unidirectional relation from the S&P 500 index to silver in PII.
Because oil, next to gold and silver, is a frequently-used investment asset as well, the causal linkages between oil and the S&P 500 index were also investi-gated. The linear Granger test results indicated unidirectional relations from oil to the S&P 500 index in PI and from the S&P 500 index to oil in PII. The DP test found a bi-directional relation between the S&P 500 index and oil in PII. It was also investigated whether oil drives the two precious metals. The five-variate im-plementation of the VAR model suggested a unidirectional linear relation from oil to gold in PI. The DP test found bi-directional nonlinear relations from oil to silver and from oil to gold in PII.
Gold’s potential role as safe haven asset suggested interdependence between gold and the exchange rate. However, little evidence was found for the existence of causal linkages between the two. Only the linear test, applied to the bivariate model, revealed a unidirectional relation from the exchange rate to gold in PI and PII. The DP test indicated a nonlinear relation from the exchange rate to the silver price in PII. Moreover, the five-variate implementation of the linear test suggested a bi-directional relation in the same period.
Only a few nonlinear relations persisted after GARCH-BEKK filtering. This means that the nonlinear causal relations can for a large part be attributed to volatil-ity effects. The five-variate model was more effective in capturing the volatilvolatil-ity effects as opposed to the pairwise models. Almost all nonlinear causality was re-moved after filtering. The relations that remained were the unidirectional relations from silver to gold in PI and PII, the unidirectional relation from the S&P 500 in-dex to silver in PI and the relation from oil to the exchange rate in PII. Causality in third or higher-order moments may be a factor of the remaining interdependencies. With use of generalized additive models, it was possible to visualize the nonlin-ear Granger causal relations found by the DP test. An interesting topic for further research is to investigate whether other specifications of the GAMs lead to dif-ferent results. In this thesis, a number of assumptions has been made. First, the GAMs consisted of two or three smooth functions and each smooth function in-volved only one covariate. Second, an additive structure was assumed. However, a variety of other potential GAMs is available. For example, a smooth function of multiple predictors may be more appropriate. Thus instead of an additive struc-ture of two smooths containing one predictor, one could estimate a GAM with one smooth containing both predictors. It is also possible to mix smooth functions and
parametric model components in one model.
Another interesting subject for further research is the choice of GARCH-BEKK model used for filtering the VAR residuals. The diagonal-BEKK(1,1,1) model cap-tured the nonlinear causal relations for a large part. However, a number of causal relations remained significant. The remaining interdependence may be the result of causality present in third or higher-order moments. It could mean however, that the GARCH model did not fit the data properly because it was incorrectly specified. Further research could point out whether other GARCH models are more effective in modeling the covariance structure.
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